Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity

Transcription

1 Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity Aleksei Netšunajev Thesis submitted for assessment with a view to obtaining the degree of Doctor of Economics of the European University Institute Florence, March 213

2 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

3 European University Institute Department of Economics Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity Aleksei Netšunajev Thesis submitted for assessment with a view to obtaining the degree of Doctor of Economics of the European University Institute Examining Board Prof. Helmut Lütkepohl, DIW Berlin and Freie Universität (External Supervisor) Prof. Fabio Canova, European University Institute Prof. Helmut Herwartz, Georg-August-Universität Göttingen Prof. Markku Lanne, University of Helsinki Aleksei Netšunajev, 213 No part of this thesis may be copied, reproduced or transmitted without prior permission of the author Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

4 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

5 ACKNOWLEDGMENTS It is a pleasure to express my appreciation to the people who have contributed to the completion of this work. I want to thank my parents, Natalja and Vladimir, and my wife Svetlana for their support and patience. I am grateful to my supervisor, Professor Helmut Lütkepohl, who encouraged and guided me through the preparation of the thesis. I am also grateful to Professors Arpad Abraham, Nicola Pavoni, Peter Hansen, and Fabio Canova. from the European University Institute for their valuable comments on parts of the thesis. I am also thankful for the comments I received during various seminars and conferences where I presented each of the papers that this thesis is comprised of. i Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

6 ABSTRACT Structural vector autoregressions are of great importance in applied macroeconometric work. The main difficulty associated with structural analysis is to identify unique shocks of interest. In a conventional approach this is done via zero or sign restrictions. Heteroskedasticity is proposed for use in identification. Under certain assumptions when volatility of shocks changes over time, unique shocks can be obtained. Then formal testing of the restrictions and impulse response analysis can be performed. In this thesis I show how identification via heteroskedasticity can be used in different contexts. In the first chapter I analyze the dynamics of trade balances in response to macroeconomic shocks. I show that identifying restrictions, which are known in the literature, are rejected for two out of seven countries. Partially identified models fail to provide enough information to fully identify shocks. The second chapter, coauthored with my supervisor, demonstrates how one can benefit from identification via heteroskedasticity when sign restrictions are used. The approach is illustrated with a model of the crude oil market. It is shown that shocks identified via previously known sign restrictions are in line with the properties of the data. Use of tighter restrictions uncovers that the approach can be discriminative. The third chapter reconsiders the conflicting results in the debate on the effects of technology shocks on hours worked. Using six ways of identifying technology shocks, I find that not all of them are supported by the data. There is no clear-cut evidence in favor of positive reaction of hours to technology shocks. However, it is plausible for real wage and disentangled investment-specific and neutral technology shocks, even though conventional identification of the latter shocks is rejected. ii Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

7 Contents Introduction v 1 Trade Balance Dynamics in Response to Macroeconomic Shocks: Conventional Identification versus Identification via Heteroskedasticity Introduction Theoretical model and conventional SVAR Model MS-VAR model with changing volatility Application to International Trade The data Model selection and statistical analysis Volatility Structure and Impulse Response Analysis Conclusions Bibliography Disentangling Demand and Supply Shocks in the Crude Oil Market: How to Check Sign Restrictions in Structural VARs Introduction The Model Setup and Inference Empirical Analysis Previous identification restrictions The data and the VAR model The number of MS states Statistical analysis of MS(3)-VAR(3) model Impulse response analysis Conclusions Bibliography iii Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

8 3 Reaction to Technology Shocks in Markov-switching Structural VARs: Identification via Heteroskedasticity Introduction Identification of shocks The Model Markov Switching SVAR Bootstrapping confidence bands Empirical analysis Statistical Analysis Impulse Response Analysis Conclusions Bibliography Appendix iv Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

9 Introduction Structural vector autoregressions (VAR) are of great importance in applied macroeconometric work. The main difficulty arising in the analysis is the identification of the structural shocks of interest. To obtain unique and economically interpretable shocks, one can usually use some theory describing the relationships of interest. Then exclusion of certain instantaneous or long-run effects on the variables, or in other words assuming zero effects helps to identify the shocks. Another way of identifying structural shocks of interest is based on sign restrictions, when one admits the whole range of shocks that are in line with the prespecified positive or negative effects. These ways of identification have a common undesirable feature, namely there is no place for the data to speak up against the restrictions. The zero restrictions just identify shocks and are not testable. The sign restrictions exclude the reaction that is not in line with the assumed effects. However, recent proposals have been made to use heteroskedasticity to extract additional identifying information. When the volatility of shocks changes over time, the sample size can be partitioned according to the changes and under certain assumptions, unique shocks are obtained. These shocks are statistically identified, yet they may not be interpretable economically. Formal testing of the restrictions and impulse response analysis reveals if the economically meaningful shocks are in line with the properties of the data. In the thesis I use the Markov switching (MS) mechanism to model and determine the changes in the volatility regimes of shocks endogenously from the data. The expectation maximization algorithm and maximum likelihood is used to estimate the models. The thesis consists of three chapters that are separate research papers in structural time series econometrics. It is shown how identification via heteroskedasticity can be used in the context of zero and sign restrictions. The methodology is illustrated with several well known examples in the literature. In the first chapter I analyze the dynamics of trade balances in response to macroeconomic shocks in the structural vector autoregressive framework. I use Markov switching vector autoregression models with volatility changes in the residuals, in combination with v Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

10 conventional procedure to identify structural shocks. Quarterly data on relative output, real exchange rates and trade balances are analyzed. The identifying restrictions are rejected for two out of seven countries. The rejection comes from the scheme imposed to identify nominal shocks. Partially identified models fail to provide enough information to fully identify shocks. Impulse response analysis is conducted for the countries for which the structural shocks can be identified. I find nominal shocks to have permanent positive effects on trade balances. Volatility analysis shows that countries are different in terms of the volatility structure. Therefore, when one is dealing with a group of countries the identification scheme may not be suitable for all of them. The second chapter was written in collaboration with Professor Helmut Lütkepohl, my thesis supervisor. My contribution to the paper is computational. I was in charge of the empirical part of the paper, namely estimation of the models, testing of the restrictions and impulse response analysis. The chapter focuses on the sign restricted structural VAR and shows how one can use identification via heteroskedasticity in the context of sign restrictions. Although in an ideal setting the sign restrictions specify shocks of interest, sign restrictions may be invalidated by measurement errors, data adjustments or omitted variables. So far there are no techniques for validating the shocks identified via such restrictions. The approach is illustrated by considering a small model for the market of crude oil. It is shown that shocks identified via sign restrictions are in line with the properties of the data. The restrictions used in the example are weak and it may be difficult to argue against them. Using similar but tighter restrictions, it is shown that the approach can be discriminative. In the third chapter I reconsider the conflicting results in the debate connected to the effects of technology shocks on hours worked. Given the considerable dissatisfaction with the just-identifying long-run restrictions, I analyze whether the restrictions used in the literature are consistent with the properties of the data. In the chapter, changes in volatility of shocks, as well as non-linearities in the intercept term are modeled with the Markov switching mechanism. Using six ways of identifying technology shock, I find that not all of them are consistent with the data. Identification of permanent technology shocks, non-permanent technology shocks and permanent real wage shocks is supported by the data. Furthermore, permanent TFP shocks and permanent technology shocks after controlling for capital tax have less support from the data. Finally, disentangling investment specific and neutral technology shocks is not supported by the data, however, a neutral technology shock can be identified in the system. There is not strong evidence in favor of positive reaction of hours to different technology shocks. But it is plausible for real wage shocks and an investment-specific neutral technology shocks tandem. The latter result is achieved even though conventional vi Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

11 identification of the shocks is rejected by the data. The identification via heteroskedasticity is a useful tool that one should not neglect in a structural VAR analysis. Combining it with the conventional approach, one can either bring convincing arguments in favour of the identified shocks or conclude that the identification scheme is not supported by the data. vii Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

12 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

13 Chapter 1 Trade Balance Dynamics in Response to Macroeconomic Shocks: Conventional Identification versus Identification via Heteroskedasticity Aleksei Netšunajev 1.1 Introduction The world economy has become increasingly integrated in recent years. The rising volume of international trade in goods as a result of contemporary organization of production schemes, when the production of single elements of a final commodity is diffused among countries is a well known phenomenon. Trade in services has also become increasingly important during the last couple of decades. All this suggests that international trade flows are subject to a range of macroeconomic shocks. Hence, understanding the reaction to shocks is essential from a number of different perspectives, including domestic and international policy coordination. In this paper I focus on trade balance adjustment that could be directly interpreted as, for instance, reaction to monetary or fiscal policy changes. The theoretical and empirical model from Prasad (1999) is used as a benchmark in the current analysis. Prasad considers relative output, real exchange rate and trade balance of the G7 1 countries in the structural vector 1 The following ISO notation is used for the countries: Canada - CA, Italy - IT, United Kingdom - GB, Germany - DE, Unites States - US, Japan - JP, France - FR 1 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

14 autoregressive (SVAR) framework to analyze the dynamics of trade balance in response to macroeconomic shocks. The econometric analysis is conducted separately for each country allowing the author to provide countrywide quantitative estimates of the effects. In the conventional structural VAR analysis as used by Prasad (1999) the major issue is to identify structural shocks. Prasad (1999) uses a strategy proposed by Blanchard and Quah (1989), where the empirical model is identified by imposing a set of long-run restrictions derived from a theoretical model. The restrictions are just-identifying and, hence, cannot be checked against the data by statistical tests. The restrictions proposed by Prasad (1999) may be subject to criticism on the following grounds. Firstly, they come from a stylized IS-LM model which may not be the best approximation of the data generation process. Hence, it is relevant to question if one should carry over the features of the models to the SVAR. Secondly, the economic implications of the restrictions might be controversial, as will be discussed later. In the recent literature statistical properties of the data have been used to extract additional identifying information. In SVAR analysis identification via heteroskedasticity has been proposed and used by Rigobon (23), Rigobon and Sack (23), and Lanne and Lütkepohl (28). The underlying assumption of those papers is that there are exogenously generated changes in the volatility of the shocks. Furthermore, one could extract the information about the volatility regimes from the data and partition the sample accordingly. Then the identification of the shocks is based on the assumption that the effects of shocks are the same regardless of the volatility regime in which they occur. In other words, the impulse responses are invariant throughout the sample period whereas the volatility of the shocks changes. Lanne and Lütkepohl (21), Lanne, Lütkepohl and Maciejowska (21) and later Herwartz and Lütkepohl (211) consider Markov switching vector error correction (VECM) models and show how residual heteroskedasticity could be used for identification purposes. As pointed out by Lanne et al. (21) analysis based solely on statistical information could produce economically meaningless and unattractive results. That means, shocks and impulse responses may provide little information on the underlying economic processes. Hence, combining conventional and statistical identifying information could be useful and give more evidence in favor of a particular identification procedure which would shed more light on the nature of economic processes. The present paper adapts the common approach of Lanne et al. (21) and Herwartz and Lütkepohl (211) and applies it to the trade balance analysis. I model the conditional heteroskedasticity of the residuals assuming Markov switching structure. Going for Markov 2 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

15 switching heteroskedasticity models has two purposes in the present study. Firstly, using specific covariance matrix parameterization I can formally test for the validity of possible restrictions and let the data speak, either in favor of or against the imposed identification structure. Secondly, it allows me to capture unobserved volatility regimes (states) of the considered empirical models and detect different behavior of the economic indicators conditional to the regimes. Estimated state probabilities and variances of the variables usually have interesting economic interpretations. One would typically associate state probabilities with specific events in the economies and pin down the reaction of the variables to regime shifts via estimated variances. Thus the paper contributes to the literature in the following ways. Firstly, recent econometric and identification methodology is applied to the structural VAR case. The study of Prasad (1999) is reconsidered and the issue of identification of the shocks is addressed, giving the data the possibility to speak up against the imposed identifying restrictions. Secondly, the results from the MS models show how G7 countries trade balance and other aggregate variables behave in different volatility periods, allowing one to see which indicators are most vulnerable to economic fluctuations. Impulse responses shed light on the response of the trade balance to shocks and transmission mechanisms of the shocks. The paper is structured as follows. In Section 1.2 the theoretical model and a SVAR with conventional identification procedure used by Prasad (1999) is shortly discussed. In Section 1.3 the MS extension of the econometric model that allows for statistical identification of the shocks is presented. Description of the data and detailed empirical analysis follows in Section 1.4. Conclusions are provided in the last section. All computations and estimations are done in Matlab. 1.2 Theoretical model and conventional SVAR Model This section briefly presents the theoretical model that illustrates the main channels through which different types of macroeconomic shocks influence the dynamics of the trade balance. The model is adapted from Prasad (1999) and it is a stochastic version of the Mundell- Fleming model that incorporates sluggish price adjustment. With the exception of the interest rate and the trade balance, the variables in the model are in logarithms and, except for the trade balance, are expressed as deviations of domestic levels from foreign levels of the corresponding variables. The model reads: y d t = d t + η(s t p t ) σ(i t E t [p t+1 p t ]) (1.1) 3 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

16 p t = (1 θ)e t 1 p e t + θp e t, < θ 1 (1.2) m s t p t = y t λi t (1.3) i t = E t [s t+1 s t ] (1.4) In the model yt d denotes output demand, d t is a demand shock, s t is the nominal exchange rate, p t is the aggregate price level, i t is the nominal interest rate, p e t is equilibrium flexible price level, m s t is money supply. Equation (1.1) is an open economy IS equation, equation (1.2) captures sluggish adjustment of the price level to its flexible price equilibrium, where θ is the speed of adjustment, equation (1.3) is an LM equation and (1.4) is an interest rate parity condition. Remember that in the national income accounting identity, GDP is the sum of domestic demand and net export of goods and nonfactor services. It is therefore sufficient to specify the determinants of the trade balance since, given total output, this accounting identity then pins down total domestic demand. The two main determinants of the trade balance are assumed to be relative output and the real exchange rate. The equation for the home country s trade balance then reads: tb t = ξq t βy t (1.5) where ξ and β denote elasticities of the trade balance with respect to real exchange rate q t and relative output y t. This specification implies that, if business cycles were perfectly synchronized between the home and foreign countries, the composition of domestic output would depend solely on the level of the real exchange rate. The stochastic processes that drive the relative supply of output, the relative real demand shock, and relative money supply are as follows (the associated shocks are supply, demand, and nominal shock). The first and the last stochastic processes are assumed to be random walks while the demand shock is allowed to have a permanent as well as a transitory component: yt s = yt 1 s + z t d t = d t 1 γδ t 1 + δ t m t = m t 1 + v t The innovations z t, δ t, v t are assumed to be serially and mutually uncorrelated. The flexible-price rational expectations solution to the model is: yt e = yt s (1.6) 4 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

17 qt e = ys t d t + σγδ t η η(η + σ) (1.7) where y e t and q e t are the equilibrium output and real exchange rate. Trade balance then could be rewritten as: tb e t = y s t ( ) ξ η β + ξ ( ) σγδt η η + σ d t (1.8) Equations (1.6) - (1.8) represent the rational expectations solution to the model. These equations imply that the level of output is not affected by either nominal or demand shocks and nominal shock does not influence the level of the real exchange rate. These are the three restrictions that were used to identify the econometric model. The restriction on the level of output might be controversial, as it may well be that the demand shock shifts output to a new level. Recalling the fact that the data in Prasad (1999) was in first differences, the theoretical restrictions translate into the so-called long run restrictions as in Blanchard and Quah (1989). One could also see that in the model the level of trade balance is not influenced by nominal shocks. The last implication may be questionable, therefore it is left unused for the identification purposes. The model indicates that the effects of supply shocks and nominal shocks on the trade balance in the short run are ambiguous and depend on the elasticities of the trade balance w.r.t. relative output and the real exchange rate. On the other hand, permanent demand shocks, which result in an appreciation of the real exchange rate simultaneously with a transitory increase in relative output, produce an unambiguous negative trade balance response. Due to permanent (random walk) components of the shocks, the model implies that relative output, real exchange rate and trade balance are non-stationary in levels but stationary in first differences. Since the dynamics are determined by different shocks, the variables are not cointegrated. These implications will be further tested against the data. The econometric methodology used to estimate the data is as follows. Consider a standard K dimensional reduced form VAR(p) model as in Lütkepohl (25): y t = v + A 1 y t A p y t p + u t (1.9) were y t = (y 1t,..., y Kt ) is a K 1 dimensional vector of observations, v is the intercept term, A i are K K coefficient matrices for i = 1,..., p and u t is a K 1 dimensional error term that has mean zero and positive definite covariance matrix u t (, Σ u ). In the standard SVAR approach based on (1.9) a transformation of the reduced form residuals u t is used to obtain the structural shocks denoted by ɛ t. A transformation matrix B is chosen such that ɛ t = B 1 u t (, I K ), i.e. structural shocks are assumed to be 5 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

18 orthogonal with variances normalized to unity. Therefore Σ u = BB. To obtain identified, unique structural shocks, one should impose restrictions on the B matrix. Zero restrictions are usually used. A zero restriction on B implies that a certain shock does not have an instantaneous effect on one of the variables. The long-run restrictions could be imposed using methodology discussed by Blanchard and Quah (1989). They consider the accumulated effect of shocks to the system, which is summarized by the total impact matrix: Ξ = (I K A 1... A p ) 1 B and they identify the structural innovations by placing zero restrictions on this matrix. For Ξ to exist, it must be that the [I K A 1... A p ] matrix is invertible, that is, the VAR process is stationary. Prasad (1999) uses this approach for the 3-dimensional (K = 3) system. Given the dimensionality one needs to impose K(K 1)/2 = 3 restrictions on the total impact matrix. The restrictions on the long-run matrix can be transformed into restrictions on the B matrix in the way shown by Lütkepohl (25). The restrictions are justified by the theoretical model discussed above, i.e. nominal and demand shocks have no effects on the level of output and that nominal shocks do not have a permanent effect on the level of the real exchange rate. Ordering the variables as (RelativeOutput t, RealExchangeRate t, T B t ) and associating supply, demand and nominal shocks with structural innovations (ɛ 1t, ɛ 2t, ɛ 3t ) respectively, the total impact matrix has three zero restrictions: Ξ =. The unrestricted elements are denoted by asterisks. Notice, that these zero restrictions are just identifying and, hence, cannot be tested with statistical methods. 1.3 MS-VAR model with changing volatility It is a well established fact that economic time series exhibit changes in their behavior throughout nearly any horizon of observations. These changes could represent such events in the economy as financial crises (Jeanne and Masson, 2) or policy changes (Hamilton, 1989, 1994). Therefore, dynamics of economic indicators in different time periods have recently been of increased research interest. The Markov switching (MS) models for analyzing changes in the time series were first discussed by Hamilton (1989). Parameters of the considered autoregression were viewed as the outcome of a discrete-state Markov process. A general econometric framework for multiple time series when the data generation process is subject to 6 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

19 regime shifts has been constructed by Krolzig (1997) as a generalization and continuation of work by Hamilton (1988, 1989, 1994). A stable vector autoregression is analyzed, conditional on the regime, and the regime generating process is assumed to follow a Markov chain. The regime switching or Markov switching (MS) models were found to be applicable and outperforming traditional models in business cycle, fiscal and monetary analysis and in the analysis of financial data. Thus, regime switching models work well in many situations where multiple time series models have traditionally been used. The way to proceed in allowing for MS in the multivariate structural VAR model is to consider the MS-VAR with heteroskedasticity, i.e different volatility regimes, a MSH-VAR in Krolzig (1997) notation. The main reasons for doing this in the current study is to account for potential heteroskedasticity and extract additional identifying information from the data. To allow for the regime change, one should think of u t from (1.9) being dependent on the discrete Markov process s t with states 1,..., M. The probabilistic model could be described as: p ij = P r(s t = j s t 1 = i), i, j = 1,..., M where p ij s are the transition probabilities from state i to j. In addition to that the conditional distribution of u t given s t is assumed to be normal: u t s t N(, Σ m ) The distributional assumption is used for setting up the likelihood function in the estimation procedure. The normality assumption is not crucial for the identification of shocks. The conditional likelihood function is: f(y t s t = m, Y t 1 ) = (2π) K/2 det(σ m ) 1/2 exp(.5u tσ 1 m u t ), m = 1...M. where matrix Y t 1 represents the history of observations up to time t. The main feature of the model is that Σ m depends on the individual state. The identification of shocks is based on the assumption that they are orthogonal across states with variances changing across states. Remember that the B matrix is the matrix of instantaneous effects, hence state invariant instantaneous effects are assumed. The assumption on the effects of structural shocks does not go beyond what is typically assumed in a conventional SVAR analysis. Orthogonality of the shocks across the sample is a standard assumption in the literature (Lütkepohl, 25; Lanne et al., 21). If there are just two states with positive definite covariance matrices Σ 1 and Σ 2, a re- 7 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

20 sult of matrix algebra establishes that there exists a K K matrix B such that Σ 1 = BB and Σ 2 = BΛ 2 B, where Λ 2 = diag(λ 1,..., λ K ) is a diagonal matrix with positive diagonal elements (Lanne et al., 21). It turns out that the decomposition of the state dependent covariance matrices is crucial for the identification procedure. Lanne et al. (21) and Herwartz and Lütkepohl (211) show how the decomposition of covariance matrices helps to test the identification restrictions on the B matrix. Consider the two state system. Decomposition Σ 1 = BB and Σ 2 = BΛ 2 B gives a total of K(K + 1) equations which can be solved uniquely for the K 2 elements of B and the K diagonal elements of Λ 2, if the diagonal elements of Λ 2 are all distinct. Therefore B is uniquely identified (up to sign changes and interchange of columns). Thus restrictions imposed on B become over-identifying and testable. The diagonal elements of the matrix Λ 2 have a nice interpretation and could be thought of as relative variances of the structural shocks in State 2 versus State 1. Sign reversal of B is not a problem. As shown by Lanne et al. (21) the matrix is still identified if all elements in any of the columns are multiplied by 1. This means to consider negative shocks instead of positive ones (or vice versa). If there are more than two volatility states, the corresponding covariance matrix decomposition: Σ 1 = BB, Σ i = BΛ i B, i = 2,..., M with diagonal Λ i may have problems with uniqueness and existence. B is unique up to sign reversal in its columns if for each pair of equal diagonal elements in, say Λ 2 = diag(λ 21,..., λ 2K ) there is a corresponding pair of distinct diagonal elements in other Λ i = diag(λ i1,..., λ ik ), i = 3,..., M. For instance, if λ 2k = λ 2l then it should be that λ ik λ il for some i = 3,..., M. (Lanne et al., 21). Assumption that the decomposition exists is restrictive. Therefore if there are more than two states, the invariance of the instantaneous effects of shocks across states could be tested by a likelihood ratio test where under the null Σ 1,..., Σ M have the decomposition discussed above. As shown by Lanne et al. (21) the corresponding likelihood ratio test has an asymptotic χ 2 distribution with.5mk(k + 1) K 2 (M 1)K degrees of freedom. If M = 2, the number of degrees of freedom is zero. Under the stated assumptions maximum likelihood (ML) estimation is the most suitable method of estimation. If normality of the conditional distribution does not hold, the estimates are pseudo ML. The estimation procedure is based on the EM algorithm as in Hamilton (1994) and Krolzig (1997) and non-linear optimization of the log likelihood as in 8 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

21 Lanne et al. (21) and Herwartz and Lütkepohl (211). Understandingly such a procedure is computationally demanding and the results may depend on the starting values. To reduce the possibility that the optimizer ends up in a local optimum the algorithm is repeated for a whole range of starting values and the estimates that correspond to the highest likelihood are chosen. 1.4 Application to International Trade The data The data collected for the current study is different from that used in Prasad (1999). The main reason for that is the willingness to broaden the time horizon and hence have more data to analyze rather than simply check and replicate the existing results. Nevertheless, the new datasets are constructed in a way that allows the capture of the same effects as in the benchmark paper. The data, apart from real exchange rates, originates from the IMF International Financial Statistics database. The source for the real exchange rate time series is the Bank for International Settlements (narrow indices). To construct the datasets I use quarterly data on GDP which is seasonally adjusted, exports of goods and services and imports of goods and services. Monthly real exchange rates were averaged to construct quarterly series. For each G7 country I construct an index of relative demand by taking a trade-weighted average of GDP in the other G7 countries. In order to derive relative output the log of this index is subtracted from the log of domestic output. I use the same weights as in the benchmark paper, they can be seen in Table 1.1. They represent the proportion of the total trade for each country accounted for by the other G7 countries. These weights are further normalized so that for each G7 country the sum of the weights of the other six countries in the sample is 1. The trade balance is measured as the difference between exports of goods Table 1.1: Trade weights for the G7 countries Canada France Germany Italy Japan U.K. U.S and services and imports of goods and services. The trade balance is expressed as a ratio of total output in order to control for scale effects which is standard practice in the literature. All variables except for the trade balance are used in logs. The following notation for the 9 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

22 variables is used throughout the paper Y - relative output (log), RER - real exchange rate (log), T B - trade balance. The data was collected for each G7 country from 1964:Q1 through 29:Q4. The two exceptions with respect to the time horizon are Germany and France. The French dataset starts from 197:Q1 and the German one from 1978:Q3 due to the availability of import and export data in the International Financial Statistics database. To see if the data preserves certain features previously reported in the literature I compute correlations of trade balance (EX-IM) and GDP of the countries (see Table 1.2). It is evident that trade balance is positively correlated with GDP for CA, IT, JP and FR. For the other countries I find negative correlations. Findings for all countries apart from GB and DE are consistent with Mendoza (1995). However it should be noted, firstly, that Mendoza (1995) applies a Hodrick-Prescott filter to the data which I don t do; secondly, I get a positive correlation for DE if I consider a time period only until 199 as the author did Model selection and statistical analysis In this section I deal with the model selection process and discuss such issues as stationarity of the series, lag order, number of states and statistical analysis of the restrictions. I consider first differenced VAR models for all countries as a suitable choice for the following reasons. First, there is strong evidence of unit roots in nearly all series with only some evidence of cointegration for one country - Japan. This will be discussed later. Second, it allows me to focus on a homogeneous sample. And third, this is in line with both the theoretical model described above and the benchmark paper. ADF tests for models with nonzero mean and trend were conducted to check for stationarity of the series. Test statistics are presented in Table 1.3. Presence of the unit root could not be rejected at 1% level for all series and all countries apart from the trade balance of Japan. To check for further evidence in favor of first differencing variables I conducted a series of cointegration tests. Neither Johansen s trace test (Johansen, 1991) nor Saikkonen and Lütkepohl s test for cointegration (Saikkonen and Lütkepohl, 2) provide evidence for cointegration relations in the data for any of the countries except for Japan. In the latter case, one could not reject the cointegration rank rk = 1 at 1% level. These results are a slight problem, as the unit root test results for Japanese trade balances implies that the data exhibits different behavior than the theoretical model. Also existing cointegration relation requires error correction specification for the Japanese data. For those reasons I drop Japan from the subsequent analysis. Furthermore in order to determine the lag order for each country I use Akaike Information Criterion (AIC) and Hannan-Quinn Criterion (HQ) values for VAR models without 1 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

23 Markov switching. Based on AIC, lag order 1 is chosen for CA, IT, GB, DE; lag order 2 for US and lag order 4 for FR data. HQ criteria generally supports the selected lag order. It should be said that the benchmark paper uses likelihood ratio tests to determine the lag order. This results in the optimal lags being In the current study however, I stick to the lags selected by AIC for two reasons. Firstly, there are few if any significant coefficients left in the subsequent lags and secondly, the optimization algorithm is sensitive to overparametrization of the model. Equipped with the results I estimate VAR models for each country. Resulting residuals are shown in Figure 1.1. One can clearly see that the volatility of the residuals changes across time for each country. This speaks in favor of the necessity to model heteroskedasticity. Choosing the number of volatility states is crucial for the analysis. Standard LR tests are not suitable for the purpose due to the fact that the usual regularity conditions are not fulfilled under the null hypothesis and some parameters are unidentified, hence the asymptotic null distribution of the likelihood ratio test statistic is not the χ 2 one. Although some special tests have been proposed in the literature they are computationally very demanding and complicated. An alternative way to choose the number of states is based on the multipleregime representation of the MS-VAR(m, p) process and determines the state dimension m by means of complexity-penalized likelihood criteria, such as AIC, HQ or Swarz Criterion (SC). The criteria have a general form C(θ) = 2 log L T (θ) + C T dim(θ) where C T is a constant that for AIC is equal to 2 and for SC is log T. Psaradakis and Spagnolo (23) examined the performance of the three procedures for selecting the number of regimes. The authors conclude that AIC is generally successful in choosing the correct state dimension, whereas HQ and SC have a tendency to underestimate the number of states. In order to select the number of states and to test for the identification restrictions I estimate two and three state restricted, partially restricted and unrestricted models with state invariant B for each of the countries. I also estimate three state model without assumptions on B and use the results to test for the validity of state invariant instantaneous effects. AIC and SC values for two and three state models with time invariant B and with/without long run (BQ) restrictions are computed for each country and reported in Table 1.4. Although there is no general agreement in the literature whether it is sufficient to use the information criteria to select number of states, it is instructive to compare models on those grounds as, for example, in Herwartz and Lütkepohl (211). It is quite clear that AIC 11 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

24 favors three state models for all countries with an exception of DE, whereas SC favors three states only for IT. This should not be a surprise as, in general, AIC chooses models with a high number of estimated parameters. If one looks at the estimated state probabilities for three state models, then, unfortunately, for the majority of the countries one of the states would be associated with relatively few periods. Hence, the estimates could be unreliable due to the small number of observations associated with one of the states. Another argument for the two state models in the current analysis could be based on the standardized residuals. Those are computed as ˆΣ 1/2 t t 1ût, where û t = y t ˆν Â1y t 1 Âpy t p and ˆΣ t t 1 = M Pr(s t = m Y t 1 )ˆΣ m. m=1 Here ˆΣ t t 1 denotes the estimated residual covariance matrix conditional on the information up to time t 1, that is, conditional on Y t 1 = (Y t 1,..., Y 1 ). The residuals for the two state models are shown in Figure 1.2. If we compare those with the VAR residuals, then the standardized ones are much more homogeneous. Therefore the MS structure with two states captures the turbulences of volatility well enough. Consequently, in what follows I continue with the two state models. The next question to address is whether the estimated B is locally unique. Even though the two state covariance matrix decomposition is unique (up to sign changes and rotation of columns) the estimation precision could be poor due to the features of the data. One should therefore take a closer look at the estimated λ ij parameters. The estimates and standard errors are presented in Table 1.5. Note that for restricted models no ordering of the elements in Λ 2 is imposed, while for the unrestricted model the elements go from smallest to largest. The estimates of λ ij s are all different, but they may not be significantly different due to high standard errors (especially for CA, DE and US). Some words on the estimation of standard errors should be said. I have used the outer product estimate of the information matrix as in Hamilton (1994). Covariance matrix of the parameters is then the inverse of the information matrix, while standard errors are obtained by taking square roots of the diagonal elements of the covariance matrix. Remembering that estimated parameters are asymptotically normally distributed, Wald and LR tests will suit for the purpose of testing the equality of the λ ij s. The Wald test uses 12 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

25 estimates of the covariance matrices. The number of estimated parameters relative to the sample size and duration of the states is quite large and therefore, the covariance matrix could be poorly estimated. High standard errors constitute the evidence of that. Therefore Wald tests may only by of limited use whereas conducting LR tests could be beneficial. The LR test depends on the highly nonlinear likelihood optimization subject to constraints. Nevertheless, LR tests are preferable in the context. For uniqueness of the B in a two state model it suffices to test whether λ 2i = λ 2j for all combinations of i and j. The corresponding Wald and LR test statistics are shown in Table 1.6. On the one hand, Wald tests clearly fail to reject equality of λ ij s for all countries apart from IT. For GB and FR two pairwise equalities are rejected. The results for the other countries reflect relatively high standard errors. On the other hand, looking at the LR statistics one could find strong evidence in favor of structural decomposition. For CA, IT and DE all null hypotheses of pairwise equality of λ ij s are rejected at most at 7% significance level. This should be interpreted as evidence for the existence of decomposition for these datasets. For GB and FR two and one pairwise equalities could not be rejected respectively. The rejection of equality comes from the estimation precision and should not be interpreted as evidence against the decomposition. As to US data, pairwise equalities could not be rejected at common significance level, the smallest p value is.261. This may be due to the features of the data, the Markov switching model may not be the best approximation of the data generation process for the dataset. The first state is represented only by a few observations (as it becomes clear from Figure 1.7), while the estimated number of parameters is high. Therefore the estimates of the B may be poor in that case. Based on LR tests there is strong evidence for the identified and unique B for all countries apart from the US. This gives good grounds to look at the LR tests for different restrictions reported in Table 1.7. Considering two state model, the restrictions imposed by Prasad (1999) are not in line with the data for two of the analyzed countries. The long run restrictions are rejected at 1% level for CA (note that the result is independent of the number of states). As to IT, GB, US and FR the restrictions for the two state case could not be rejected at least at 8% level, thus the data speaks quite strongly in favor of those. For DE restrictions are rejected at a low level for both two and three state models, but the data objects to the restrictions less in the three state model. The data for CA and DE did not support the restrictions and in the data for JP cointegration was present. Thus, the chosen structural model may not be the best one to apply to those countries data. Looking at the information criteria, SC favors long run restrictions for GB, US and FR, whereas AIC always chooses models with neither assumptions on the B matrix nor long run restrictions. 13 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498

26 Therefore SC values are consistent with the LR test results. In order to analyze the reasons for the rejection of the restrictions, I estimate partially identified models for CA and DE. The estimated models are those with the identified demand shock ξ 1,2 =, combination of the restrictions ξ 1,2 =, ξ 2,3 = and ξ 1,2 =, ξ 1,3 = and the identified nominal shock ξ 1,3 =, ξ 2,3 =. As becomes clear, the data does not object to the identification of the demand shock, but the identification of the nominal shock is problematic. For CA the model with maximum two restrictions ξ 1,2 =, ξ 2,3 = is supported, while for DE only ξ 1,2 = could not be rejected at a common significance level. This means that the relations in the equations (1.6) and (1.7) may not be valid for all countries. In other words, the demand shocks are identified for CA and DE, but the other two may not have clear economic interpretation. This issue will be reconsidered in the impulse response analysis section. Summing up, independently of the number of states the long run restrictions were rejected for CA and DE. However the model where only demand shocks are identified for those countries was supported by the data. On the other hand, sufficient support for the long run restrictions in the two state case was found for IT, GB, US and FR Volatility Structure and Impulse Response Analysis Some joint features of the volatility structure could be seen among the countries. Estimated covariance matrices for the datasets are reported in Table 1.8, smoothed state probabilities are presented in Figures Trade balance of the European countries and CA has the lowest variance in any of the volatility states, followed by RER. Relative output is the most volatile variable for these countries. Low volatility of trade balance could be an indicator of a very similar and interconnected trade structure among the countries. This would not be a surprise for the European Union countries who have been members of a customs union since 1958, but the data from Canada also speaks in favor of trade volatility structure similarity to EU countries. On the other hand the US have an exactly reversed feature. Trade balance is the variable with the highest variance in the high volatility states. Although relative output of the US is also highly volatile in the state that was previously associated with economic recessions, it is not as volatile as the trade balance. It could be partly due to the fact that the US is the largest economy in the sample and reaction to changed economic and trade conditions is different if compared to relatively small EU countries. The US economy is also known to be highly sensitive to the prices of raw material (mainly crude oil), that form a substantial part in traded goods. Another feature to note for the US is that RER is the variable with the 14 Netšunajev, Aleksei (213), Structural Vector Autoregressions with Markov Switching: Identification via Heteroskedasticity European University Institute DOI: 1.287/73498